Asymptotics for a trigonometric integral

[jozko.slaninka]

I have become stuck while trying to evaluate the following trigonometric integral:
$\int_0^{\pi} (A + B \sin x)^n dx.$
First, I have tried to find a recurrence with respect to n, from which the closed-form solution could be calculated. However, I have failed to do this. Similarly, my effort to solve the integral using the binomial theorem seems to be useless, since this leads to the sum that I find relatively hard to evaluate.

So my next thoughts have been to find an asymptotic approximation of this integral, for n tending to infinity. However, I have found that I am unable to do this with my very basic background in asymptotic analysis.

Thus, my question is: could anybody suggest me how to achieve the above mentioned asymptotic approximation (I don't believe in a reasonable closed-form solution of this integral, so I am not asking for any)?

Thank you in advance.

 

[jvicens]

Thank you for sharing your question with us. Evaluating trigonometric integrals can be a challenging task, especially when the exponent is variable. I would like to offer some suggestions that may help you in finding an asymptotic approximation for this integral.

Firstly, it is important to note that the integrand contains a trigonometric function raised to a power. This suggests that we may be able to use Euler's formula, which states that e^{ix} = cosx + isinx, to rewrite the integral in terms of complex exponentials. This can often simplify the integration process.

Secondly, you mentioned trying to find a recurrence relation for the integral with respect to n. This is a good approach, as it may help us to identify any patterns or relationships that could lead to a solution. One way to do this is by using integration by parts, which can help to reduce the power of the trigonometric term.

Lastly, you mentioned your basic background in asymptotic analysis. I would recommend looking into the method of steepest descent, which is commonly used to find asymptotic approximations for integrals with large exponents. This method involves finding the saddle points of the integrand and using them to deform the integration contour.

I hope these suggestions are helpful in your pursuit of an asymptotic approximation for this integral. Keep exploring different approaches and don't hesitate to seek help from fellow scientists or mathematics experts. Best of luck in your research.